Divided Difference

The divided difference , sometimes also denoted  (Abramowitz and Stegun 1972), on  points , , ...,  of a function  is defined by  and

(1)

for . The first few differences are

			(2)
			(3)
			(4)

Defining

(5)

and taking the derivative

(6)

gives the identity

(7)

Consider the following question: does the property

(8)

for  and  a given function guarantee that  is a polynomial of degree ? Aczél (1985) showed that the answer is "yes" for , and Bailey (1992) showed it to be true for  with differentiable . Schwaiger (1994) and Andersen (1996) subsequently showed the answer to be "yes" for all  with restrictions on  or .

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972.

Aczél, J. "A Mean Value Property of the Derivative of Quadratic Polynomials--Without Mean Values and Derivatives." Math. Mag. 58, 42-45, 1985.

Andersen, K. M. "A Characterization of Polynomials." Math. Mag. 69, 137-142, 1996.

Bailey, D. F. "A Mean-Value Property of Cubic Polynomials--Without Mean Values." Math. Mag. 65, 123-124, 1992.

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 439-440, 1987.

Jeffreys, H. and Jeffreys, B. S. "Divided Differences." §9.012 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 260-264, 1988.

Schwaiger, J. "On a Characterization of Polynomials by Divided Differences." Aequationes Math. 48, 317-323, 1994.

Sauer, T. and Xu, Y. "On Multivariate Lagrange Interpolation." Math. Comput. 64, 1147-1170, 1995.

Whittaker, E. T. and Robinson, G. "Divided Differences" and "Theorems on Divided Differences." §11-12 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 20-24, 1967.